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advanced algorithms
advanced algorithms

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... – It is easy to merge two lists of numbers, each of which is already sorted, into a single sorted list – So: divide the list into two equal parts, sort each part with some method, then merge the two sorted lists into a single sorted list – … actually, to sort each of the parts, we can again use Merg ...
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... If a prime appears in both lists, choose it and remove it from both lists and repeat the process until there are no common elements. The product of all the chosen numbers is the GCD. Question: How old is this algorithm? Example: 948 and 156 948 = 2 x 2 x 3 x 79 156 = 2 x 2 x 3 x 13 Common prime fact ...
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Finite fields - CSE

< 1 ... 210 211 212 213 214 215 216 217 218 ... 231 >

Factorization of polynomials over finite fields

In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors. This decomposition is theoretically possible and is unique for polynomials with coefficients in any field, but rather strong restrictions on the field of the coefficients are needed to allow the computation of the factorization by means of an algorithm. In practice, algorithms have been designed only for polynomials with coefficients in a finite field, in the field of rationals or in a finitely generated field extension of one of them.The case of the factorization of univariate polynomials over a finite field, which is the subject of this article, is especially important, because all the algorithms (including the case of multivariate polynomials over the rational numbers), which are sufficiently efficient to be implemented, reduce the problem to this case (see Polynomial factorization). It is also interesting for various applications of finite fields, such as coding theory (cyclic redundancy codes and BCH codes), cryptography (public key cryptography by the means of elliptic curves), and computational number theory.As the reduction of the factorization of multivariate polynomials to that of univariate polynomials does not have any specificity in the case of coefficients in a finite field, only polynomials with one variable are considered in this article.
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